In
mathematics, the Appell–Humbert theorem describes the
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
s on a
complex torus
In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', w ...
or complex
abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functi ...
.
It was proved for 2-dimensional tori by and , and in general by
Statement
Suppose that
is a complex torus given by
where
is a lattice in a complex vector space
. If
is a Hermitian form on
whose imaginary part
is integral on
, and
is a map from
to the unit circle
, called a semi-character, such that
:
then
:
is a 1-
cocycle
In mathematics a cocycle is a closed cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likewise used in group cohomology. In autonomous d ...
of
defining a line bundle on
. For the trivial Hermitian form, this just reduces to a
character
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
. Note that the space of character morphisms is isomorphic with a real torus
if
since any such character factors through
composed with the exponential map. That is, a character is a map of the form
for some covector
. The periodicity of
for a linear
gives the isomorphism of the character group with the real torus given above. In fact, this torus can be equipped with a complex structure, giving the
dual complex torus.
Explicitly, a line bundle on
may be constructed by
descent
Descent may refer to:
As a noun Genealogy and inheritance
* Common descent, concept in evolutionary biology
* Kinship, one of the major concepts of cultural anthropology
** Pedigree chart or family tree
**Ancestry
**Lineal descendant
** Heritage ...
from a line bundle on
(which is necessarily trivial) and a
descent data, namely a compatible collection of isomorphisms
, one for each
. Such isomorphisms may be presented as nonvanishing holomorphic functions on
, and for each
the expression above is a corresponding holomorphic function.
The Appell–Humbert theorem says that every line bundle on
can be constructed like this for a unique choice of
and
satisfying the conditions above.
Ample line bundles
Lefschetz proved that the line bundle
, associated to the Hermitian form
is ample if and only if
is positive definite, and in this case
is very ample. A consequence is that the complex torus is algebraic if and only if there is a positive definite Hermitian form whose imaginary part is integral on
See also
*
Complex torus
In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', w ...
for a treatment of the theorem with examples
References
*
*
*
*
*
{{DEFAULTSORT:Appell-Humbert theorem
Abelian varieties
Theorems in algebraic geometry
Theorems in complex geometry